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On permutation invariance and nonseparability of N-qubit states

Ting Gao, Fengli Yan, S. J. van Enk

Published Jan 18, 2014• 0 citations
OpenAlex

Abstract

We consider concept of the permutationally invariant (PI) part of a density matrix, which has proven very useful for both efficient quantum state estimation and entanglement characterization of $N$-qubit systems. We show here that concept is, in fact, basis-dependent, but that this basis dependence makes it an even more powerful concept than has been appreciated so far. By considering PI part of a general (mixed) $N$-qubit state, we obtain: (i) strong bounds on quantitative nonseparability measures, (ii) a whole hierarchy of multi-partite separability criteria (one of which entails a sufficient criterion for genuine $N$-partite entanglement) that can be experimentally determined by just $2N+1$ measurement settings, (iii) a definition of an efficiently measurable degree of separability, which can be used for quantifying a novel aspect of decoherence of $N$ qubits, and (iv) an explicit example that shows there are, for increasing $N$, genuinely $N$-partite entangled states lying closer and closer to maximally mixed state. We further argue to add as requirement on any multi-partite entanglement measure $E$ that it satisfy $E(\rho)\geq E(\rhopi)$, even though operation that maps $\rho\rightarrow\rhopi$ is not local.

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We consider concept of the permutationally invariant (PI) part of a density matrix, which has proven very useful for both efficient quantum state estimation and entanglement characterization of NNN-qubit systems. We show here that concept is, in fact, basis-dependent, but that this basis dependence makes it an even more powerful concept than has been appreciated so far. By considering PI part of a general (mixed) NNN-qubit state, we obtain: (i) strong bounds on quantitative nonseparability measures, (ii) a whole hierarchy of multi-partite separability criteria (one of which entails a sufficient criterion for genuine NNN-partite entanglement) that can be experimentally determined by just 2N+12N+12N+1 measurement settings, (iii) a definition of an efficiently measurable degree of separability, which can be used for quantifying a novel aspect of decoherence of NNN qubits, and (iv) an explicit example that shows there are, for increasing NNN, genuinely NNN-partite entangled states lying closer and closer to maximally mixed state. We further argue to add as requirement on any multi-partite entanglement measure EEE that it satisfy E(ρ)≥E(\rhopi)E(\rho)\geq E(\rhopi)E(ρ)≥E(\rhopi), even though operation that maps ρ→\rhopi\rho\rightarrow\rhopiρ→\rhopi is not local.
We consider the concept of "the permutationally invariant (PI) part of a density matrix," which has proven very useful for both efficient quantum state estimation and entanglement characterization of N-qubit systems. We show here that the concept is, in fact, basis dependent but that this basis dependence makes it an even more powerful concept than has been appreciated so far. By considering the PI part ρ(PI) of a general (mixed) N-qubit state ρ, we obtain (i) strong bounds on quantitative nonseparability measures, (ii) a whole hierarchy of multipartite separability criteria (one of which entails a sufficient criterion for genuine N-partite entanglement) that can be experimentally determined by just 2N+1 measurement settings, (iii) a definition of an efficiently measurable degree of separability, which can be used for quantifying a novel aspect of decoherence of N qubits, and (iv) an explicit example that shows there are, for increasing N, genuinely N-partite entangled states lying closer and closer to the maximally mixed state. Moreover, we show that if the PI part of a state is k nonseparable, then so is the actual state. We further argue to add as requirement on any multipartite entanglement measure E that it satisfy E(ρ)≥E(ρ(PI)), even though the operation that maps ρ→ρ(PI) is not local.
We investigate the detection of entanglement in nnn-partite quantum states. We obtain practical separability criteria to identify genuinely entangled and nonseparable mixed quantum states. No numerical optimization or eigenvalue evaluation is needed, and our criteria can be evaluated by simple computations involving components of the density matrix. We provide examples in which our criteria perform better than all known separability criteria. Specifically, we are able to detect genuine nnn-partite entanglement which has previously not been identified. In addition, our criteria can be used in today's experiments.
The kkk-separability of mixed nnn-partite (n≥3n\geq 3n≥3) quantum states is mainly discussed in this paper. We obtain practical kkk-separability criteria to identify kkk-nonseparable mixed multipartite states in arbitrary dimensional quantum systems and detect genuine nnn-partite entanglement (k=2k=2k=2) and entanglement (k=3k=3k=3) which have previously not been identified. No optimization or eigenvalue evaluation is needed, our criteria can be evaluated by simple computations involving components of the density matrix and can be implemented in today's experiments.
Detection of kkk-nonseparable nnn-partite quantum states
The detection of multipartite entanglement in arbitrary dimensional systems is investigated. We derive useful kkk-separability criteria of mixed nnn-partite (n≥3n\geq 3n≥3) quantum states to detect kkk-nonseparable nnn-partite quantum states. Our criteria can be expressed by the elements of the density matrix, which allows a simple and practical evaluation and computation. They are experimentally accessible without quantum state tomography.
Many-body quantum systems can be characterized using the notions of kkk-separability and entanglement depth. A quantum state is kkk-separable if it can be expressed as a mixture of kkk entangled subsystems, and its entanglement depth is given by the size of the largest entangled subsystem. In this paper we propose a multipartite entanglement measure that satisfies the following criteria: (i) It can be used with both pure and mixed states; (ii) it is encoded in a single element of the density matrix, so it does not require knowledge of the full spectrum of the density matrix; (iii) it can be applied to large systems; and (iv) it can be experimentally verified. The proposed method allows the certification of non-kkk-separability of a given quantum state. We show that the proposed method successfully classifies three-qubit systems into known stochastic local operations and classical communication (SLOCC) classes, namely, bipartite, WWW-, and Greenberger-Horne-Zeilinger-type entanglement. Furthermore, we characterize the non-kkk-separability in nine known SLOCC classes of four-qubit states and absolutely maximally entangled states for five and six qubits and for arbitrary-size-qubit Dicke states.
In this work we consider the permutational properties of multipartite entanglement monotones. Based on the fact that genuine multipartite entanglement is a property of the entire multiqubit system, we argue that ideal definitions for its characterizing quantities must be permutation invariant. Using this criterion, we examine the three four-qubit entanglement monotones introduced by Osterloh and Siewert [Phys. Rev. A 72, 012337 (2005)]. By expressing them in terms of quantities whose permutational properties can be easily derived, we find that one of these monotones is not permutation invariant. We propose a permutation-invariant entanglement monotone to replace it, and show that our new monotone properly measures the genuine four-qubit entanglement in four-qubit cluster-class states. Our results provide some useful insights in understanding multipartite entanglement.